Intro to Group Theory | Hacker News Reader

Intro to Group Theory

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62 pointsprofquail17 comments

My AI doctorate used a lot of group theory. I use came up with the following analogy for explaining what it is and what it's used for to fellow non-mathematicians:

If you want to measure and describe the number of occurrences of something, use an integer. If you want to measure and describe the amount of something, use a rational number. If you want to measure and describe the symmetry of a structure, use a group.

When programmatically dealing with objects that contain symmetry - e.g. physical objects like a cube, a set of identical resources like a fleet of lorries - having an understanding of group theory allows you another layer of abstraction in your computation, potentially greatly reducing runtime or memory consumption.

Groups are just the abstraction of "transformations of something", the usual example being "symmetries" but it leaves out a bunch of other natural ideas like translations, movements, etc.

Any teacher not mentioning "shape" when explaining group theory is either a bad teacher or a bad teacher.

Group theory can be taught and appreciated without the geometric approach, but instead as a pure abstract concept. Different textbooks reflect this: Herstein is very abstract, Dummit and Foote uses probably more geometric examples, and Fraleigh relies heavily on geometric cases. I've personally always preferred the Herstein/abstract approach.

Oh certainly: that is the way I studied and enjoyed it but a book is not a teacher. I am not claiming books should always be "imagination" driven, on the contrary. But teachers should.

Wow.. I think this is the first time I actually get what means group theory. Excellent analogy! That was a very elusive concept to me when I took an abstract algebra course back in my undergraduate life. Thanks!

And I never had a prof who didn't describe groups as encoding symmetry. Strange.

I launched into a rant about this in a recent blog post[1]. The maths lecturer I had said, "A group theorist aims to classify and categorise the whole world..." and then he went straight into writing the four axioms of a group.

[1] http://lifebeyondfife.com/the-difference-between-mean-and-me...

Mathematics student here, taken a few courses in abstract algebra. Group theory is really interesting and I'd recommend all those intrigued to give it a go.

Here is a great book about Abstract Algebra. It should be about right for this course and it's free! :D http://abstract.ups.edu/

For anyone interested in another great way of looking into this field, take a look at Aluffi's Algebra Chapter 0[1].

[1] http://www.mimuw.edu.pl/~jarekw/pdf/Algebra0TextboookAluffi....

Hello, I'm the UReddit admin and author of the linked post. I'm glad to see appreciation here for Dr. Donley's class.

As we say in the article, we are trying to put together a better platform for teachers to use when they want to teach with more freedom and with more rights to their intellectual property rights than they might have when doing it through a university, and so that they can have many parts of the teaching process automated in order to be able to focus on actually playing the roles of educators.

We're running a Kickstarter to help us get our prototype into beta, and we appreciate any and all support. http://www.kickstarter.com/projects/1918344721/open-compass

I haven't heard of UReddit before.

What's the pitch?

UReddit was a proof of concept based on which we founded the nonprofit. The pitch is essentially that anyone that would like to teach should be free to do so, so we made a place where people can do that. Now, Dr. Donley, who taught the class the linked blog post is about, had been composing video lectures for two years, but he did use the UReddit platform to run two classes and receive more attention/recognition for his efforts and quality of execution. And now we'd like to make a technologically sophisticated platform that automated the busy work of teaching and is better suited to becoming a community.

I'm taking an Introductory Group Theory course right now.

Here are a few nice books I'm currently using:

http://www.amazon.com/Introduction-Theory-Groups-Joseph-Rotm...

http://www.amazon.com/Visual-Classroom-Resource-Materials-Pr...

http://www.amazon.com/Groups-Their-Graphs-Mathematical-Libra... (available as eBook from http://www.maa.org/ebooks/nml/NML14.html)

Group theory is a beautiful subject, as is the rest of abstract algebra. My personal favorite result in mathematics is the Fundamental Theorem of Galois Theory, which requires at least some understanding of group theory.

And interesting enough, you can use the same machinery developed for those problems to prove the impossibility of squaring the circle, or trisecting an angle.

Looking forward to this. Group Theory has always evaded me.

I'm in math grad school, and I'm really surprised CS majors don't know this stuff or find it useful. Probably this should be changed.

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